Bake Mesh Analysis to Texture
There is not such feature for now, but ... specifically speaking about Thickness there is an OSL shader by Róbert László Páli, which use the same type of measurement - depth along the normal of the surface.
Mesh Analysis - Thickness
with values Minimum (Red) = 0, Maximmum (Blue) = 1

OSL Shader
To set the same distance use Map Range node From Min = 0, From Max = 1



Thanks to Robin Betts for pointing me to Róbert's thread.